{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### 一.原理介绍\n", "主成分分析想做的事如下图所示，将原始坐标系$$变换为新的坐标系$$，是的原始数据在新的坐标系下具有如下特点： \n", "\n", "（1）在第一个坐标轴上的投影分散的尽可能的开（第一主成分），在第二个坐标轴上的投影其次（第二主成分）.... \n", "\n", "（2）新的坐标轴彼此正交，这样可以去掉变量间的相关性，如下面第一张图中数据几乎排在$y=x$这条线上，而经过pca的变换后，数据各维度失去了相关性\n", "![avatar](./source/19_pca1.png)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 二.求解第一个坐标轴（第一主成分）\n", "\n", "接下来让我们先从第一个坐标轴入手，根据我们的目标，我们要让原始的数据点$X\\in R^{m\\times n}$在某一个方向$w\\in R^n$上的投影点尽可能的分散，这里涉及两个点： \n", "\n", "（1）如果计算投影坐标； \n", "\n", "（2）如何衡量分散程度； \n", "\n", "下面依次推导\n", "#### 投影坐标的计算\n", "\n", "抽样坐标的计算推导如下图，将某样本点$x$看做一个向量，将坐标轴方向向量$w$移动到与$x$相同的原点，假设两向量的夹角为$\\theta$，在$w^Tw=1$的约束条件下，经过推导可得投影坐标可表示为$x^Tw$\n", "![avatar](./source/19_pca2.png)\n", "\n", "#### 分散程度的衡量：方差\n", "\n", "对于分散程度，显然最直接的一个参考指标可以使用方差，方差越大说明数据越分散，那么在坐标轴$w$上所有数据投影点的方差如下： \n", "\n", "$$\n", "S(X,w)=\\sum_{i=1}^m(x_i^Tw-\\bar{x}^Tw)^2\n", "$$ \n", "\n", "这里，$x_i\\in R^n$，$\\bar{x}=\\frac{\\sum_{i=1}^m x_i}{m}$，通常为了方便可以先进行**去中心化**的操作，即令$X$中的样本点都减去它每个维度的均值\n", "\n", "$$\n", "X=X-\\bar{X}\n", "$$ \n", "\n", "所以这时的方差就可以简单写作： \n", "\n", "$$\n", "S(X,w)=\\sum_{i=1}^m(x_i^Tw)^2=\\sum_{i=1}^m w^Tx_ix_i^Tw=w^T\\sum_{i=1}^m (x_ix_i^T)w=w^T[x_1,x_2,...,x_m]\\cdot\\begin{bmatrix}\n", "x_1^T\\\\ \n", "x_2^T\\\\ \n", "...\\\\ \n", "x_m^T\n", "\\end{bmatrix}w=w^TX^TXw\n", "$$ \n", "\n", "所以，我们我们要求最优$w^*$便是 \n", "\n", "$$\n", "w^*=arg\\max_{w}w^TX^TXw\n", "$$ \n", "\n", "这依赖于$X^TX$的性质" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 三.基于$X^TX$求解前$k$个主成分\n", "\n", "由于$X^TX$为实对称矩阵，我们可以对其进行正交分解，且其特征值均大于等于0，所以我们可以将$X^TX$分解为如下形式： \n", "\n", "$$\n", "X^TX=U\\Lambda U^T=[u_1,u_2,...,u_n]\\begin{bmatrix}\n", "\\lambda_1 & 0 & 0 & 0 \\\\ \n", "0 & \\lambda_2 & 0 & 0\\\\ \n", "0 & 0 & \\ddots & 0\\\\ \n", "0 & 0 & 0 & \\lambda_n\n", "\\end{bmatrix}\\begin{bmatrix}\n", "u_1^T\\\\ \n", "u_2^T\\\\ \n", "...\\\\ \n", "u_n^T\n", "\\end{bmatrix}\n", "$$ \n", "\n", "其中，$\\lambda_1\\geq\\lambda_2\\geq\\cdots\\geq\\lambda_n\\geq 0,u_i^Tu_j=\\left\\{\\begin{matrix}\n", "1 &i=j \\\\ \n", "0 &i\\neq j\n", "\\end{matrix}\\right.\n", "$\n", "\n", "对于，任意$w(s.t. w^Tw=1)$，都有： \n", "\n", "$$\n", "w^TX^TXw=(w^Tu_1)^2\\lambda_1+(w^Tu_2)^2\\lambda_2+\\cdots+(w^Tu_n)^2\\lambda_n\n", "$$ \n", "\n", "\n", "由于$u_1,u_2,...,u_n$彼此正交，且为单位向量，所以$(w^Tu_1)^2+(w^Tu_2)^2+\\cdots+(w^Tu_n)^2=1$，所以结果也就可以看出来了： \n", "\n", "（1）对于第一个坐标轴（第一主成分），即是$\\lambda_1$对应的特征向量$u_1$...\n", "\n", "（2）对于第二个坐标轴（第二主成分），即是$\\lambda_2$对应的特征向量$u_2$...\n", "\n", "（3）...... \n", "\n", "通常，我们选择前$k$个坐标轴进行映射$U_k=[u_1,u_2,...,u_k]$，生成的新数据为$X_k=XU_k$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 四.代码实现\n", "\n", "#### 造伪数据\n", "将数据点$(x,y)$分布在$y=3x+2$上下" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "#造伪样本\n", "X=np.linspace(0,10,100)\n", "Y=3*X+2\n", "X+=np.random.normal(size=(X.shape))*0.3#添加噪声\n", "data=np.c_[X,Y]" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0,0.5,'y')" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "plt.scatter(data[:,0],data[:,1])\n", "plt.xlabel('x')\n", "plt.ylabel('y')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 去中心化" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "data=data-np.mean(data,axis=0,keepdims=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 求$X^TX$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "xTx=data.T.dot(data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 正交分解" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "eig_vals,eig_vecs=np.linalg.eig(xTx)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([8.27256772e+00, 8.50444256e+03])" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eig_vals" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[-0.94847792, -0.31684323],\n", " [ 0.31684323, -0.94847792]])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eig_vecs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "校验一下特征分解结果" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 1.25055521e-12, 0.00000000e+00],\n", " [ 0.00000000e+00, -9.09494702e-13]])" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eig_vecs.dot(np.diag(eig_vals)).dot(eig_vecs.T)-xTx" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "由于`np.linalg.eig`的特征值并未按从大到小的顺序排序，所以我们需要进行一下重排" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "sorted_indice=np.argsort(-1*eig_vals)#默认是升序排序\n", "eig_vals=eig_vals[sorted_indice]\n", "eig_vecs[:]=eig_vecs[:,sorted_indice]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在的特征矩阵，第一列便是第一主成分，第二列便是第二主成分，接下来我们可以轻易拿到转换后的数据，只需要去中心化数据矩阵和特征矩阵相乘即可" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "new_data=data.dot(eig_vecs)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 变化后的数据" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-20, 20)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(new_data[:,0],new_data[:,1])\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.xlim(-20, 20)\n", "plt.ylim(-20, 20)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "可以发现，最终得到了我们想要的结果...，最后将代码整理一下，放到`ml_models.decomposition.PCA`中" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 2 }